Package org.opengis.referencing.cs Description

Coordinate system

The coordinates of points are recorded in a coordinate system. A coordinate
system is the set of coordinate system axes that spans the coordinate space. This concept implies
the set of mathematical rules that determine how coordinates are associated with invariant
quantities such as angles and distances. In other words, a coordinate system implies how
coordinates are calculated from geometric elements such as distances and angles and vice versa.
The calculus required to derive angles and distances from point coordinates and vice versa in a
map plane is simple Euclidean 2D arithmetic. To do the same on the surface of an ellipsoid
(curved 2D space) involves more complex ellipsoidal calculus. These rules cannot be specified in
detail, but are implied in the geometric properties of the coordinate space.

NOTE: The word "distances" is used loosely in
the above description. Strictly speaking distances are not invariant quantities, as they are
expressed in the unit of measure defined for the coordinate system; ratios of distances are
invariant.

One coordinate system
may be used by multiple coordinate reference systems. A coordinate system is composed of an ordered set of coordinate
system axes, the number of axes being equal to the dimension of the space of which it describes
the geometry. Its axes can be spatial, temporal, or mixed. Coordinates in coordinate tuples shall
be supplied in the same order as the coordinate axes are defined.

The dimension of the coordinate space, the names, the units of measure, the
directions and sequence of the axes are all part of the Coordinate System definition. The number
of coordinates in a tuple and consequently the number of coordinate axes in a coordinate system
shall be equal to the number of coordinate axes in the coordinate system. It is therefore not
permitted to supply a coordinate tuple with two heights of different definition in the same
coordinate tuple.

Coordinate systems are divided into subtypes by the geometric properties of
the coordinate space spanned and the geometric properties of the axes themselves (straight or
curved; perpendicular or not). Certain subtypes of coordinate system can only be used with
specific subtypes of coordinate reference system.

Coordinate system axis

A coordinate system is composed of an ordered set of coordinate system axes.
Each of its axes is completely characterised by a unique combination of axis name, axis
abbreviation, axis direction and axis unit of measure.

The concept of coordinate axis requires some clarification. Consider an
arbitrary x, y, z coordinate system. The x-axis may
be defined as the locus of points with y = z = 0. This
is easily enough understood if the x, y, z coordinate system is
a Cartesian system and the space it describes is Euclidean. It becomes a bit more difficult to
understand in the case of a strongly curved space, such as the surface of an ellipsoid, its
geometry described by an ellipsoidal coordinate system (2D or 3D). Applying the same definition
by analogy to the curvilinear latitude and longitude coordinates the latitude axis would be the
equator and the longitude axis would be the prime meridian, which is not a satisfactory
definition.

Bearing in mind that the order of the coordinates in a coordinate tuple must
be the same as the defined order of the coordinate axes, the "i-th" coordinate axis of
a coordinate system is defined as the locus of points for which all coordinates with sequence
number not equal to "i", have a constant value locally (whereby i =
1...n, and n is the dimension of the coordinate space).

It will be evident that the addition of the word "locally" in this definition
apparently adds an element of ambiguity and this is intentional. However, the definition of the
coordinate parameter associated with any axis must be unique. The coordinate axis itself should
not be interpreted as a unique mathematical object, the associated coordinate parameter should.
For example, geodetic latitude is defined as the "Angle from the equatorial plane to the
perpendicular to the ellipsoid through a given point, northwards usually treated as positive".
However, when used in an ellipsoidal coordinate system the geodetic latitude axis will be
described as pointing "north". In two different points on the ellipsoid the direction "north"
will be a spatially different direction, but the concept of latitude is the same.

Furthermore the specified direction of the coordinate axes is often only
approximate; two geographic coordinate reference systems will make use of the same ellipsoidal
coordinate system. These coordinate systems are associated with the earth through two different
geodetic datums, which may lead to the two systems being slightly rotated w.r.t. each other.